Another consequence of the scalar self-interaction is level-mixing, which may shut down some superradiant modes. We focus on the effect of the dominant \(\lbrace 0,1,1\rbrace \)  mode on the subdominant \(\lbrace 1,1,1\rbrace \)  mode. The self-interaction could annihilate two \(\lbrace 1,1,1\rbrace \)  scalars into a scalar in the \(\lbrace 0,1,1\rbrace \)  mode and a scalar with energy \(\omega ^{\prime }=2\omega _R^{111}-\omega _R^{011}\)  and \(m^{\prime }=1\) . With the approximation of \(\omega _R\)  in Eq. (REF ), one gets \(\omega ^{\prime }>\mu \) , implying it is a continuous mode. Following the same argument as in Ref. [1]}, the flux at the horizon is,
\(\omega _R^{011}\left(\omega _R^{011} - \Omega _H \right) \left|u_h\right|^2+\omega ^{\prime }\left(\omega ^{\prime } - \Omega _H \right) \left|v_h\right|^2,\) 
